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Properties

Label 882.2.h.q.79.3
Level $882$
Weight $2$
Character 882.79
Analytic conductor $7.043$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(67,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,0,-4,0,0,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.3
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 882.79
Dual form 882.2.h.q.67.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(1.22474 - 1.22474i) q^{3} +(-0.500000 + 0.866025i) q^{4} -3.86370 q^{5} +(-1.67303 - 0.448288i) q^{6} +1.00000 q^{8} -3.00000i q^{9} +(1.93185 + 3.34607i) q^{10} -3.73205 q^{11} +(0.448288 + 1.67303i) q^{12} +(3.34607 + 5.79555i) q^{13} +(-4.73205 + 4.73205i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(2.70831 + 4.69093i) q^{17} +(-2.59808 + 1.50000i) q^{18} +(-1.48356 + 2.56961i) q^{19} +(1.93185 - 3.34607i) q^{20} +(1.86603 + 3.23205i) q^{22} -1.46410 q^{23} +(1.22474 - 1.22474i) q^{24} +9.92820 q^{25} +(3.34607 - 5.79555i) q^{26} +(-3.67423 - 3.67423i) q^{27} +(2.00000 - 3.46410i) q^{29} +(6.46410 + 1.73205i) q^{30} +(0.896575 - 1.55291i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-4.57081 + 4.57081i) q^{33} +(2.70831 - 4.69093i) q^{34} +(2.59808 + 1.50000i) q^{36} +(-0.267949 + 0.464102i) q^{37} +2.96713 q^{38} +(11.1962 + 3.00000i) q^{39} -3.86370 q^{40} +(-0.637756 - 1.10463i) q^{41} +(-1.86603 + 3.23205i) q^{43} +(1.86603 - 3.23205i) q^{44} +11.5911i q^{45} +(0.732051 + 1.26795i) q^{46} +(5.27792 + 9.14162i) q^{47} +(-1.67303 - 0.448288i) q^{48} +(-4.96410 - 8.59808i) q^{50} +(9.06218 + 2.42820i) q^{51} -6.69213 q^{52} +(1.46410 + 2.53590i) q^{53} +(-1.34486 + 5.01910i) q^{54} +14.4195 q^{55} +(1.33013 + 4.96410i) q^{57} -4.00000 q^{58} +(-4.31199 + 7.46859i) q^{59} +(-1.73205 - 6.46410i) q^{60} +(3.48477 + 6.03579i) q^{61} -1.79315 q^{62} +1.00000 q^{64} +(-12.9282 - 22.3923i) q^{65} +(6.24384 + 1.67303i) q^{66} +(-2.76795 + 4.79423i) q^{67} -5.41662 q^{68} +(-1.79315 + 1.79315i) q^{69} +2.53590 q^{71} -3.00000i q^{72} +(-3.41542 - 5.91567i) q^{73} +0.535898 q^{74} +(12.1595 - 12.1595i) q^{75} +(-1.48356 - 2.56961i) q^{76} +(-3.00000 - 11.1962i) q^{78} +(2.46410 + 4.26795i) q^{79} +(1.93185 + 3.34607i) q^{80} -9.00000 q^{81} +(-0.637756 + 1.10463i) q^{82} +(-8.95215 + 15.5056i) q^{83} +(-10.4641 - 18.1244i) q^{85} +3.73205 q^{86} +(-1.79315 - 6.69213i) q^{87} -3.73205 q^{88} +(3.53553 - 6.12372i) q^{89} +(10.0382 - 5.79555i) q^{90} +(0.732051 - 1.26795i) q^{92} +(-0.803848 - 3.00000i) q^{93} +(5.27792 - 9.14162i) q^{94} +(5.73205 - 9.92820i) q^{95} +(0.448288 + 1.67303i) q^{96} +(-2.94855 + 5.10703i) q^{97} +11.1962i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 16 q^{11} - 24 q^{15} - 4 q^{16} + 8 q^{22} + 16 q^{23} + 24 q^{25} + 16 q^{29} + 24 q^{30} - 4 q^{32} - 16 q^{37} + 48 q^{39} - 8 q^{43} + 8 q^{44} - 8 q^{46} - 12 q^{50}+ \cdots + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 1.22474 − 1.22474i 0.707107 − 0.707107i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −3.86370 −1.72790 −0.863950 − 0.503577i \(-0.832017\pi\)
−0.863950 + 0.503577i \(0.832017\pi\)
\(6\) −1.67303 − 0.448288i −0.683013 − 0.183013i
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) − 3.00000i − 1.00000i
\(10\) 1.93185 + 3.34607i 0.610905 + 1.05812i
\(11\) −3.73205 −1.12526 −0.562628 − 0.826710i \(-0.690210\pi\)
−0.562628 + 0.826710i \(0.690210\pi\)
\(12\) 0.448288 + 1.67303i 0.129410 + 0.482963i
\(13\) 3.34607 + 5.79555i 0.928032 + 1.60740i 0.786612 + 0.617448i \(0.211833\pi\)
0.141420 + 0.989950i \(0.454833\pi\)
\(14\) 0 0
\(15\) −4.73205 + 4.73205i −1.22181 + 1.22181i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.70831 + 4.69093i 0.656861 + 1.13772i 0.981424 + 0.191853i \(0.0614497\pi\)
−0.324562 + 0.945864i \(0.605217\pi\)
\(18\) −2.59808 + 1.50000i −0.612372 + 0.353553i
\(19\) −1.48356 + 2.56961i −0.340353 + 0.589509i −0.984498 − 0.175395i \(-0.943880\pi\)
0.644145 + 0.764903i \(0.277213\pi\)
\(20\) 1.93185 − 3.34607i 0.431975 − 0.748203i
\(21\) 0 0
\(22\) 1.86603 + 3.23205i 0.397838 + 0.689076i
\(23\) −1.46410 −0.305286 −0.152643 − 0.988281i \(-0.548779\pi\)
−0.152643 + 0.988281i \(0.548779\pi\)
\(24\) 1.22474 − 1.22474i 0.250000 − 0.250000i
\(25\) 9.92820 1.98564
\(26\) 3.34607 − 5.79555i 0.656217 − 1.13660i
\(27\) −3.67423 − 3.67423i −0.707107 − 0.707107i
\(28\) 0 0
\(29\) 2.00000 − 3.46410i 0.371391 − 0.643268i −0.618389 − 0.785872i \(-0.712214\pi\)
0.989780 + 0.142605i \(0.0455477\pi\)
\(30\) 6.46410 + 1.73205i 1.18018 + 0.316228i
\(31\) 0.896575 − 1.55291i 0.161030 − 0.278912i −0.774209 − 0.632931i \(-0.781852\pi\)
0.935238 + 0.354019i \(0.115185\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) −4.57081 + 4.57081i −0.795676 + 0.795676i
\(34\) 2.70831 − 4.69093i 0.464471 − 0.804488i
\(35\) 0 0
\(36\) 2.59808 + 1.50000i 0.433013 + 0.250000i
\(37\) −0.267949 + 0.464102i −0.0440506 + 0.0762978i −0.887210 − 0.461366i \(-0.847360\pi\)
0.843159 + 0.537664i \(0.180693\pi\)
\(38\) 2.96713 0.481332
\(39\) 11.1962 + 3.00000i 1.79282 + 0.480384i
\(40\) −3.86370 −0.610905
\(41\) −0.637756 − 1.10463i −0.0996008 − 0.172514i 0.811919 − 0.583771i \(-0.198423\pi\)
−0.911519 + 0.411257i \(0.865090\pi\)
\(42\) 0 0
\(43\) −1.86603 + 3.23205i −0.284566 + 0.492883i −0.972504 − 0.232887i \(-0.925183\pi\)
0.687938 + 0.725770i \(0.258516\pi\)
\(44\) 1.86603 − 3.23205i 0.281314 − 0.487250i
\(45\) 11.5911i 1.72790i
\(46\) 0.732051 + 1.26795i 0.107935 + 0.186949i
\(47\) 5.27792 + 9.14162i 0.769863 + 1.33344i 0.937637 + 0.347617i \(0.113009\pi\)
−0.167773 + 0.985826i \(0.553658\pi\)
\(48\) −1.67303 − 0.448288i −0.241481 − 0.0647048i
\(49\) 0 0
\(50\) −4.96410 − 8.59808i −0.702030 − 1.21595i
\(51\) 9.06218 + 2.42820i 1.26896 + 0.340016i
\(52\) −6.69213 −0.928032
\(53\) 1.46410 + 2.53590i 0.201110 + 0.348332i 0.948886 − 0.315618i \(-0.102212\pi\)
−0.747776 + 0.663951i \(0.768879\pi\)
\(54\) −1.34486 + 5.01910i −0.183013 + 0.683013i
\(55\) 14.4195 1.94433
\(56\) 0 0
\(57\) 1.33013 + 4.96410i 0.176180 + 0.657511i
\(58\) −4.00000 −0.525226
\(59\) −4.31199 + 7.46859i −0.561373 + 0.972327i 0.436004 + 0.899945i \(0.356394\pi\)
−0.997377 + 0.0723823i \(0.976940\pi\)
\(60\) −1.73205 − 6.46410i −0.223607 − 0.834512i
\(61\) 3.48477 + 6.03579i 0.446179 + 0.772804i 0.998133 − 0.0610700i \(-0.0194513\pi\)
−0.551955 + 0.833874i \(0.686118\pi\)
\(62\) −1.79315 −0.227730
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.9282 − 22.3923i −1.60355 − 2.77742i
\(66\) 6.24384 + 1.67303i 0.768564 + 0.205936i
\(67\) −2.76795 + 4.79423i −0.338159 + 0.585708i −0.984086 − 0.177690i \(-0.943137\pi\)
0.645928 + 0.763399i \(0.276471\pi\)
\(68\) −5.41662 −0.656861
\(69\) −1.79315 + 1.79315i −0.215870 + 0.215870i
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 − 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) −3.41542 − 5.91567i −0.399744 − 0.692377i 0.593950 − 0.804502i \(-0.297568\pi\)
−0.993694 + 0.112125i \(0.964234\pi\)
\(74\) 0.535898 0.0622969
\(75\) 12.1595 − 12.1595i 1.40406 − 1.40406i
\(76\) −1.48356 − 2.56961i −0.170176 − 0.294754i
\(77\) 0 0
\(78\) −3.00000 − 11.1962i −0.339683 − 1.26771i
\(79\) 2.46410 + 4.26795i 0.277233 + 0.480182i 0.970696 − 0.240310i \(-0.0772492\pi\)
−0.693463 + 0.720492i \(0.743916\pi\)
\(80\) 1.93185 + 3.34607i 0.215988 + 0.374101i
\(81\) −9.00000 −1.00000
\(82\) −0.637756 + 1.10463i −0.0704284 + 0.121986i
\(83\) −8.95215 + 15.5056i −0.982626 + 1.70196i −0.330583 + 0.943777i \(0.607245\pi\)
−0.652043 + 0.758182i \(0.726088\pi\)
\(84\) 0 0
\(85\) −10.4641 − 18.1244i −1.13499 − 1.96586i
\(86\) 3.73205 0.402437
\(87\) −1.79315 − 6.69213i −0.192246 − 0.717472i
\(88\) −3.73205 −0.397838
\(89\) 3.53553 − 6.12372i 0.374766 − 0.649113i −0.615526 − 0.788116i \(-0.711056\pi\)
0.990292 + 0.139003i \(0.0443898\pi\)
\(90\) 10.0382 − 5.79555i 1.05812 − 0.610905i
\(91\) 0 0
\(92\) 0.732051 − 1.26795i 0.0763216 − 0.132193i
\(93\) −0.803848 − 3.00000i −0.0833551 − 0.311086i
\(94\) 5.27792 − 9.14162i 0.544376 − 0.942886i
\(95\) 5.73205 − 9.92820i 0.588096 − 1.01861i
\(96\) 0.448288 + 1.67303i 0.0457532 + 0.170753i
\(97\) −2.94855 + 5.10703i −0.299379 + 0.518540i −0.975994 − 0.217797i \(-0.930113\pi\)
0.676615 + 0.736337i \(0.263446\pi\)
\(98\) 0 0
\(99\) 11.1962i 1.12526i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.2.h.q.79.3 8
3.2 odd 2 2646.2.h.t.667.4 8
7.2 even 3 882.2.f.q.295.3 yes 8
7.3 odd 6 882.2.e.s.655.4 8
7.4 even 3 882.2.e.s.655.1 8
7.5 odd 6 882.2.f.q.295.2 ✓ 8
7.6 odd 2 inner 882.2.h.q.79.2 8
9.4 even 3 882.2.e.s.373.1 8
9.5 odd 6 2646.2.e.q.1549.1 8
21.2 odd 6 2646.2.f.r.883.1 8
21.5 even 6 2646.2.f.r.883.4 8
21.11 odd 6 2646.2.e.q.2125.1 8
21.17 even 6 2646.2.e.q.2125.4 8
21.20 even 2 2646.2.h.t.667.1 8
63.2 odd 6 7938.2.a.ci.1.4 4
63.4 even 3 inner 882.2.h.q.67.4 8
63.5 even 6 2646.2.f.r.1765.4 8
63.13 odd 6 882.2.e.s.373.4 8
63.16 even 3 7938.2.a.cp.1.1 4
63.23 odd 6 2646.2.f.r.1765.1 8
63.31 odd 6 inner 882.2.h.q.67.1 8
63.32 odd 6 2646.2.h.t.361.4 8
63.40 odd 6 882.2.f.q.589.2 yes 8
63.41 even 6 2646.2.e.q.1549.4 8
63.47 even 6 7938.2.a.ci.1.1 4
63.58 even 3 882.2.f.q.589.3 yes 8
63.59 even 6 2646.2.h.t.361.1 8
63.61 odd 6 7938.2.a.cp.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.1 8 9.4 even 3
882.2.e.s.373.4 8 63.13 odd 6
882.2.e.s.655.1 8 7.4 even 3
882.2.e.s.655.4 8 7.3 odd 6
882.2.f.q.295.2 ✓ 8 7.5 odd 6
882.2.f.q.295.3 yes 8 7.2 even 3
882.2.f.q.589.2 yes 8 63.40 odd 6
882.2.f.q.589.3 yes 8 63.58 even 3
882.2.h.q.67.1 8 63.31 odd 6 inner
882.2.h.q.67.4 8 63.4 even 3 inner
882.2.h.q.79.2 8 7.6 odd 2 inner
882.2.h.q.79.3 8 1.1 even 1 trivial
2646.2.e.q.1549.1 8 9.5 odd 6
2646.2.e.q.1549.4 8 63.41 even 6
2646.2.e.q.2125.1 8 21.11 odd 6
2646.2.e.q.2125.4 8 21.17 even 6
2646.2.f.r.883.1 8 21.2 odd 6
2646.2.f.r.883.4 8 21.5 even 6
2646.2.f.r.1765.1 8 63.23 odd 6
2646.2.f.r.1765.4 8 63.5 even 6
2646.2.h.t.361.1 8 63.59 even 6
2646.2.h.t.361.4 8 63.32 odd 6
2646.2.h.t.667.1 8 21.20 even 2
2646.2.h.t.667.4 8 3.2 odd 2
7938.2.a.ci.1.1 4 63.47 even 6
7938.2.a.ci.1.4 4 63.2 odd 6
7938.2.a.cp.1.1 4 63.16 even 3
7938.2.a.cp.1.4 4 63.61 odd 6